RC Chakraborty 03/03 to 08/3, 2007, Lecture 15 to 22 (8 hrs) Slides 1 to 79 myreaders , http://myreaders.wordpress.com/ , rcchak@gmail.com (Revised – Feb. 02, 2008)

Artificial Intelligence

Knowledge Representation

Issues, Predicate Logic, Rules

Topics

(Lectures 15, 16, 17, 18, 19, 20, 21, 22 8 hours)

Slides

1. Knowledge Representation

Introduction – KR model, typology, relationship, framework,

mapping, forward & backward representation, system

requirements; KR schemes - relational, inheritable,

inferential, declarative, procedural; KR issues - attributes,

relationship, granularity.

03-26

2. KR Using Predicate Logic

Logic representation, Propositional logic - statements,

variables, symbols, connective, truth value, contingencies,

tautologies, contradictions, antecedent, consequent,

argument; Predicate logic - expressions, quantifiers,

formula; Representing “IsA” and “Instance” relationships,

computable functions and predicates; Resolution.

27-47

3. KR Using Rules

Types of Rules - declarative, procedural, meta rules;

Procedural verses declarative knowledge & language; Logic

programming – characteristics, Statement, language, syntax

& terminology, simple & structured data objects, Program

Components - clause, predicate, sentence, subject;

Programming paradigms - models of computation, imperative

model, functional model, logic model; Forward & backward

reasoning - chaining, conflict resolution; Control knowledge.

48-77

4. References 78-79

02

Knowledge Representation

Issues, Predicate Logic, Rules

How do we represent what we know ? • Knowledge is a general term.

An answer to the question, "how to represent knowledge", requires an

analysis to distinguish between knowledge “how” and knowledge “that”.

■ knowing "how to do something".

e.g. "how to drive a car" is Procedural knowledge. ■ knowing "that something is true or false".

e.g. "that is the speed limit for a car on a motorway" is Declarative

knowledge.

• knowledge and Representation are distinct entities that play a central

but distinguishable roles in intelligent system.

■ Knowledge is a description of the world.

It determines a system's competence by what it knows. ■ Representation is the way knowledge is encoded.

It defines the performance of a system in doing something.

• Different types of knowledge require different kinds of representation.

The Knowledge Representation models/mechanisms are often based on:

◊ Logic ◊ Rules ◊ Frames ◊ Semantic Net

• Different types of knowledge require different kinds of reasoning.

03

KR -Introduction 1. Introduction

Knowledge is a general term.

Knowledge is a progression that starts with data which is of limited utility.

By organizing or analyzing the data, we understand what the data means,

and this becomes information.

The interpretation or evaluation of information yield knowledge.

An understanding of the principles embodied within the knowledge is wisdom.

• Knowledge Progression

Fig 1 Knowledge Progression

■ Data is viewed as collection of

disconnected facts.

: Example : It is raining.

■ Information emerges when

relationships among facts are

established and understood;

Provides answers to "who",

"what", "where", and "when".

: Example : The temperature dropped 15

degrees and then it started raining.

■ Knowledge emerges when

relationships among patterns

are identified and understood;

Provides answers as "how" .

: Example : If the humidity is very high

and the temperature drops substantially,

then atmospheres is unlikely to hold the

moisture, so it rains.

■ Wisdom is the pinnacle of

understanding, uncovers the

principles of relationships that

describe patterns.

Provides answers as "why" .

: Example : Encompasses understanding

of all the interactions that happen

between raining, evaporation, air

currents, temperature gradients,

changes, and raining.

04

Data Information KnowledgeOrganizing

Analyzing

Interpretation

EvaluationWisdom

Understanding

Principles

KR -Introduction

• Knowledge Model (Bellinger 1980)

The model tells, that as the degree of “connectedness” and “understanding”

increase, we progress from data through information and knowledge to

wisdom.

Fig. Knowledge Model

The model represents transitions and understanding.

the transitions are from data, to information, to knowledge, and finally to

wisdom;

the understanding support the transitions from one stage to the next

stage.

The distinctions between data, information, knowledge, and wisdom are not

very discrete. They are more like shades of gray, rather than black and

white (Shedroff, 2001).

data and information deal with the past; they are based on the gathering

of facts and adding context.

knowledge deals with the present that enable us to perform.

wisdom deals with the future, acquire vision for what will be, rather than

for what is or was.

05

Data

Information

Knowledge

Wisdom

Understandingrelations

Understandingpatterns

Understanding principles

Degree of Understanding

Degree of Connectedness

KR -Introduction • Knowledge Type

Knowledge is categorized into two major types: Tacit and Explicit.

term “Tacit” corresponds to informal or implicit type of knowledge,

term “Explicit” corresponds to formal type of knowledge. Tacit knowledge Explicit knowledge

◊ Exists within a human being; it is embodied.

◊ Exists outside a human being; it is embedded.

◊ Difficult to articulate formally.

◊ Can be articulated formally.

◊ Difficult to share/communicate.

◊ Can be shared, copied, processed and stored.

◊ Hard to steal or copy. ◊ Easy to steal or copy

◊ Drawn from experience, action, subjective insight.

◊ Drawn from artifact of some type as principle, procedure, process, concepts.

[The next slide explains more about tacit and explicit knowledge.] 06

KR -Introduction ■ Knowledge typology map

The map shows that, Tacit knowledge comes from experience, action,

subjective insight and Explicit knowledge comes from principle,

procedure, process, concepts, via transcribed content or artifact of some

type.

Fig. Knowledge Typology Map

◊ Facts : are data or instance that are specific and unique.

◊ Concepts : are class of items, words, or ideas that are known by a

common name and share common features.

◊ Processes : are flow of events or activities that describe how things work

rather than how to do things.

◊ Procedures : are series of step-by-step actions and decisions that result

in the achievement of a task.

◊ Principles : are guidelines, rules, and parameters that govern;

principles allow to make predictions and draw implications;

principles are the basic building blocks of theoretical models (theories).

These artifacts are used in the knowledge creation process to create two

types of knowledge: declarative and procedural explained below. 07

Knowledge

Tacit Knowledge

Experience

Subjective Insight

Doing (action)

ExplicitKnowledge

ProcedurePrinciples

Concept

Process

Information

Context

Data

Facts

KR -Introduction

• Knowledge Type

Cognitive psychologists sort knowledge into Declarative and Procedural

category and some researchers added Strategic as a third category.

Procedural knowledge Declarative knowledge ◊ examples : procedures, rules,

strategies, agendas, models.

◊ example : concepts, objects, facts,

propositions, assertions, semantic

nets, logic and descriptive models.

◊ focuses on tasks that must be

performed to reach a particular

objective or goal.

◊ refers to representations of objects

and events; knowledge about facts

and relationships;

◊ Knowledge about "how to do

something"; e.g., to determine if

Peter or Robert is older, first find

their ages.

◊ Knowledge about "that something

is true or false". e.g., A car has

four tyres; Peter is older than

Robert;

Note :

About procedural knowledge, there is some disparity in views.

− One view is, that it is close to Tacit knowledge; it manifests itself in the doing of

some-thing yet cannot be expressed in words; e.g., we read faces and moods.

− Another view is, that it is close to declarative knowledge; the difference is that

a task or method is described instead of facts or things.

All declarative knowledge are explicit knowledge; it is knowledge that can

be and has been articulated.

The strategic knowledge is thought as a subset of declarative knowledge. 08

KR -Introduction • Relationship among knowledge type

The relationship among explicit, implicit, tacit, declarative and procedural

knowledge are illustrated below.

Fig. Relationship among types of knowledge The Figure shows, declarative knowledge is tied to "describing" and

procedural knowledge is tied to "doing."

− The arrows connecting explicit with declarative and tacit with procedural,

indicate the strong relationships exist among them.

− The arrow connecting declarative and procedural indicates that we often

develop procedural knowledge as a result of starting with declarative

knowledge. i.e., we often "know about" before we "know how".

Therefore, we may view,

− all procedural knowledge as tacit, and

− all declarative knowledge as explicit. 09

Implicit

Doing Procedural DeclarativeDescribing

Explicit Tacit

Has been articulated

Can not bearticulated

Motor Skill(Manual)

Facts and things

Tasks and methods

Mental Skill

KnowledgeStart

No Yes

NoYes

KR -framework

1.1 Framework of Knowledge Representation (Poole 1998) Computer requires a well-defined problem description to process and also

provide well-defined acceptable solution.

To collect fragments of knowledge we need : first to formulate description

in our spoken language and then represent it in formal language so that

computer can understand. The computer can then use an algorithm to

compute an answer. This process is illustrated below.

Fig. Knowledge Representation Framework

The steps are

− The informal formalism of the problem takes place first.

− It is then represented formally and the computer produces an output.

− This output can then be represented in a informally described solution

that user understands or checks for consistency.

Note : The Problem solving requires

− formal knowledge representation, and

− conversion of informal (implicit) knowledge to formal (explicit) knowledge.

10

Compute

Problem

OutputRepresentation

Solution

Interpret

Solve

Represent

Formal

Informal

KR - framework

• Knowledge and Representation

Problem solving requires large amount of knowledge and some

mechanism for manipulating that knowledge.

The Knowledge and the Representation are distinct entities, play a

central but distinguishable roles in intelligent system.

− Knowledge is a description of the world;

it determines a system's competence by what it knows.

− Representation is the way knowledge is encoded;

it defines the system's performance in doing something.

In simple words, we :

− need to know about things we want to represent , and

− need some means by which things we can manipulate.

‡ Objects - facts about objects in the domain.

‡ Events - actions that occur in the domain.

‡ Performance - knowledge about how to do things

◊ know things to represent

‡ Meta-knowledge

- knowledge about what we know

◊ need means to

manipulate

‡ Requires some formalism - to what we represent ;

Thus, knowledge representation can be considered at two levels :

(a) knowledge level at which facts are described, and

(b) symbol level at which the representations of the objects, defined in

terms of symbols, can be manipulated in the programs.

Note : A good representation enables fast and accurate access to

knowledge and understanding of the content. 11

KR - framework • Mapping between Facts and Representation

Knowledge is a collection of “facts” from some domain.

We need a representation of facts that can be manipulated by a program.

Normal English is insufficient, too hard currently for a computer program to

draw inferences in natural languages. Thus some symbolic representation

is necessary.

Therefore, we must be able to map "facts to symbols" and "symbols to

facts" using forward and backward representation mapping.

Example : Consider an English sentence

Facts Representations

◊ Spot is a dog A fact represented in English sentence

◊ dog (Spot) Using forward mapping function the above

fact is represented in logic

◊ ∀ x : dog(x) → hastail (x)

A logical representation of the fact that

"all dogs have tails"

Now using deductive mechanism we can generate a new

representation of object :

◊ hastail (Spot) A new object representation

◊ Spot has a tail

[it is new knowledge]

Using backward mapping function to

generate English sentence

12

Facts Internal

Representation

English Representation

English understanding

English generation

Reasoning programs

KR - framework

■ Forward and Backward representation

The forward and backward representations are elaborated below :

‡ The doted line on top indicates the abstract reasoning process that a

program is intended to model. ‡ The solid lines on bottom indicates the concrete reasoning process

that the program performs.

13

Initial Facts

Internal Representation

English Representation

Forward representation mapping

Backward representation

mapping

Desired real reasoning Final

Facts

Operated by program

KR - framework • KR System Requirements

A good knowledge representation enables fast and accurate access to

knowledge and understanding of the content.

A knowledge representation system should have following properties.

◊ Representational Adequacy

The ability to represent all kinds of knowledge that

are needed in that domain.

◊ Inferential Adequacy The ability to manipulate the representational

structures to derive new structure corresponding to

new knowledge inferred from old .

◊ Inferential Efficiency The ability to incorporate additional information into

the knowledge structure that can be used to focus the

attention of the inference mechanisms in the most

promising direction.

◊ Acquisitional

Efficiency The ability to acquire new knowledge using automatic

methods wherever possible rather than reliance on

human intervention.

Note : To date no single system can optimizes all of the above properties.

14

KR - schemes 1.2 knowledge Representation schemes

There are four types of Knowledge representation - Relational, Inheritable,

Inferential, and Declarative/Procedural.

◊ Relational Knowledge :

− provides a framework to compare two objects based on equivalent

attributes.

− any instance in which two different objects are compared is a relational

type of knowledge. ◊ Inheritable Knowledge

− is obtained from associated objects.

− it prescribes a structure in which new objects are created which may inherit

all or a subset of attributes from existing objects.

◊ Inferential Knowledge

− is inferred from objects through relations among objects.

− e.g., a word alone is a simple syntax, but with the help of other words in

phrase the reader may infer more from a word; this inference within

linguistic is called semantics.

◊ Declarative Knowledge

− a statement in which knowledge is specified, but the use to which that

knowledge is to be put is not given.

− e.g. laws, people's name; these are facts which can stand alone, not

dependent on other knowledge;

Procedural Knowledge

− a representation in which the control information, to use the knowledge, is

embedded in the knowledge itself.

− e.g. computer programs, directions, and recipes; these indicate specific

use or implementation;

These KR schemes are detailed below in next few slides

15

KR - schemes • Relational knowledge : associates elements of one domain with another.

Used to associate elements of one domain with the elements of another

domain or set of design constrains.

− Relational knowledge is made up of objects consisting of attributes and

their corresponding associated values.

− The results of this knowledge type is a mapping of elements among

different domains.

The table below shows a simple way to store facts.

− The facts about a set of objects are put systematically in columns.

− This representation provides little opportunity for inference.

Table - Simple Relational Knowledge

Player Height Weight Bats - Throws

Aaron 6-0 180 Right - Right

Mays 5-10 170 Right - Right

Ruth 6-2 215 Left - Left

Williams 6-3 205 Left - Right

‡ Given the facts it is not possible to answer simple question such as :

" Who is the heaviest player ? ".

‡ But if a procedure for finding heaviest player is provided, then these

facts will enable that procedure to compute an answer.

16

KR - schemes

• Inheritable knowledge : elements inherit attributes from their parents.

The knowledge is embodied in the design hierarchies found in the

functional, physical and process domains. Within the hierarchy, elements

inherit attributes from their parents, but in many cases, not all attributes of

the parent elements be prescribed to the child elements.

− The basic KR needs to be augmented with inference mechanism, and

− Inheritance is a powerful form of inference, but not adequate.

The KR in hierarchical structure, shown below, is called “semantic network”

or a collection of “frames” or “slot-and-filler structure". It shows property

inheritance and way for insertion of additional knowledge.

− Property inheritance : Objects/elements of specific classes inherit attributes

and values from more general classes.

− Classes are organized in a generalized hierarchy.

Fig. Inheritable knowledge representation (KR)

‡ the directed arrows represent attributes (isa, instance, and team) originating

at the object being described and terminating at the object or its value.

‡ the box nodes represents objects and values of the attributes. [Continuing in the next slide] 17

Person Right

Adult Male

Baseball Player

Brooklyn-Dodger

Pee-Wee-Reese

Three Finger Brown

Chicago Cubs

Pitcher Fielder

5.10

6.1 bats

isa

0.2620.106

EQUAL handed

height

instance

batting-average

team

isa isa

isa

0.252

batting-average

handed

height

instance

team

batting-average

Baseball knowledge

− isa : show class inclusion

− instance : show class membership

KR - schemes [Continuing from previous slide – example]

◊ Viewing a node as a frame

Baseball-player

isa : Adult-Male

Bates : EQUAL handed

Height : 6.1

Batting-average : 0.252

◊ Algorithm : Property Inheritance

Retrieve a value V for an attribute A of an instance object O.

Steps to follow: 1. Find object O in the knowledge base.

2. If there is a value for the attribute A then report that value.

3. Else, see if there is a value for the attribute instance; If not, then fail.

4 Else, move to the node corresponding to that value and look for a value for the attribute A; If one is found, report it.

5. Else, do until there is no value for the “isa” attribute or until an answer is found :

(a) Get the value of the “isa” attribute and move to that node.

(b) See if there is a value for the attribute A; If yes, report it.

This algorithm is simple,

‡ It does describe the basic mechanism of inheritance.

‡ It does not say what to do if there is more than one value of the instance or “isa” attribute.

This can be applied to the example of knowledge base illustrated to

derive answers to the following queries :

− team (Pee-Wee-Reese) = Brooklyn–Dodger

− batting–average(Three-Finger-Brown) = 0.106

− height (Pee-Wee-Reese) = 6.1

− bats(Three Finger Brown) = right

[For explanation - refer book on AI by Elaine Rich & Kevin Knight, page 112]

18

KR - schemes • Inferential knowledge : generates new information .

Generates new information from the given information. This new

information does not require further data gathering form source, but does

require analysis of the given information to generate new knowledge.

− Given a set of relations and values, one may infer other values or relations.

− In addition to algebraic relations, a predicate logic (mathematical deduction)

is used to infer from a set of attributes.

− Inference through predicate logic uses a set of logical operations to relate

individual data. The symbols used for the logic operations are :

" → " (implication), " ¬ " (not), " V " (or), " Λ " (and),

" ∀ " (for all), " ∃ " (there exists).

Examples of predicate logic statements :

1. Wonder is a name of a dog : dog (wonder)

2. All dogs belong to the class of animals : ∀ x : dog (x) → animal(x)

3. All animals either live on land or in water : ∀ x : animal(x) → live (x, land) V live (x, water)

We can infer from these three statements that :

" Wonder lives either on land or on water."

As more information is made available about these objects and their

relations, more knowledge can be inferred. 19

KR - schemes • Declarative/Procedural knowledge

The difference between Declarative/Procedural knowledge is not very clear.

Declarative knowledge :

Here, the knowledge is based on declarative facts about axioms and

domains.

− axioms are assumed to be true unless a counter example is found to

invalidate them.

− domains represent the physical world and the perceived functionality.

− axiom and domains thus simply exists and serve as declarative

statements that can stand alone.

Procedural knowledge:

Here the knowledge is a mapping process between domains that specifies

“what to do when” and the representation is of “how to make it” rather than

“what it is”. The procedural knowledge :

− may have inferential efficiency, but no inferential adequacy and

acquisitional efficiency.

− are represented as small programs that know how to do specific things,

how to proceed.

Example : a parser in a natural language has the knowledge that a noun

phrase may contain articles, adjectives and nouns. It thus accordingly call

routines that know how to process articles, adjectives and nouns. 20

KR - issues 1.3 Issues in Knowledge Representation

The fundamental goal of Knowledge Representation is to facilitate

inferencing (conclusions) from knowledge. The issues that arise while using

KR techniques are many. Some of these are explained below.

◊ Important Attributes : Any attribute of objects so basic that they occur

in almost every problem domain ?

◊ Relationship among attributes: Any important relationship that exists

among object attributes ?

◊ Choosing Granularity : At what level of detail should the knowledge be

represented ?

◊ Set of objects : How sets of objects be represented ?

◊ Finding Right structure : Given a large amount of knowledge stored,

how can relevant parts be accessed ?

Note : These issues are briefly explained, referring previous example, Fig.

Inheritable KR. For detail readers may refer book on AI by Elaine Rich &

Kevin Knight- page 115 – 126. 21

KR - issues • Important Attributes : Ref. Example- Fig. Inheritable KR

There are two attributes "instance" and "isa", that are of general

significance. These attributes are important because they support property

inheritance.

• Relationship among attributes : Ref. Example- Fig. Inheritable KR

The attributes we use to describe objects are themselves entities that we

represent. The relationship between the attributes of an object,

independent of specific knowledge they encode, may hold properties like:

Inverses , existence in an isa hierarchy , techniques for reasoning about

values and single valued attributes.

◊ Inverses :

This is about consistency check, while a value is added to one attribute. The

entities are related to each other in many different ways. The figure shows

attributes (isa, instance, and team), each with a directed arrow, originating at

the object being described and terminating either at the object or its value.

There are two ways of realizing this:

‡ first, represent both relationships in a single representation; e.g., a logical

representation, team(Pee-Wee-Reese, Brooklyn–Dodgers), that can be

interpreted as a statement about Pee-Wee-Reese or Brooklyn–Dodger.

‡ second, use attributes that focus on a single entity but use them in pairs,

one the inverse of the other; for e.g., one, team = Brooklyn–Dodgers ,

and the other, team = Pee-Wee-Reese, . . . .

This second approach is followed in semantic net and frame-based systems,

accompanied by a knowledge acquisition tool that guarantees the consistency

of inverse slot by checking, each time a value is added to one attribute then

the corresponding value is added to the inverse.

22

KR - issues ◊ Existence in an isa hierarchy :

This is about generalization-specialization, like, classes of objects and

specialized subsets of those classes, there are attributes and specialization

of attributes. Example, the attribute height is a specialization of general

attribute physical-size which is, in turn, a specialization of physical-attribute.

These generalization-specialization relationships are important for attributes

because they support inheritance.

◊ Techniques for reasoning about values :

This is about reasoning values of attributes not given explicitly. Several

kinds of information are used in reasoning, like,

height : must be in a unit of length,

age : of person can not be greater than the age of person's parents.

The values are often specified when a knowledge base is created.

◊ Single valued attributes :

This is about a specific attribute that is guaranteed to take a unique value.

Example, a baseball player can at time have only a single height and be a

member of only one team. KR systems take different approaches to provide

support for single valued attributes.

23

KR - issues • Choosing Granularity

Regardless of the KR formalism, it is necessary to know :

− At what level should the knowledge be represented and what are the

primitives ?."

− Should there be a small number or should there be a large number of

low-level primitives or High-level facts.

− High-level facts may not be adequate for inference while Low-level

primitives may require a lot of storage.

Example of Granularity :

− Suppose we are interested in following facts:

John spotted Sue.

− This could be represented as

Spotted (agent(John), object (Sue))

− Such a representation would make it easy to answer questions such are :

Who spotted Sue ?

− Suppose we want to know :

Did John see Sue ?

− Given only one fact, we cannot discover that answer.

− We can add other facts, such as

Spotted (x , y) → saw (x , y)

− We can now infer the answer to the question.

24

KR - issues • Set of objects

There are certain properties of objects that are true as member of a

set but not as individual;

Example : Consider the assertion made in the sentences :

"there are more sheep than people in Australia", and

"English speakers can be found all over the world."

To describe these facts, the only way is to attach assertion to the sets

representing people, sheep, and English.

The reason to represent sets of objects is : If a property is true for all or

most elements of a set, then it is more efficient to associate it once with the

set rather than to associate it explicitly with every elements of the set . This

is done,

− in logical representation through the use of universal quantifier, and

− in hierarchical structure where node represent sets and inheritance propagate set level assertion down to individual.

However in doing so, for example: assert large (elephant), remember to

make clear distinction between,

− whether we are asserting some property of the set itself, means, the set of elephants is large, or

− asserting some property that holds for individual elements of the set ,

means, any thing that is an elephant is large. There are three ways in which sets may be represented by.

(a) Name, as in the example – Fig. Inheritable KR, the node - Baseball-Player and

the predicates as Ball and Batter in logical representation.

(b) Extensional definition is to list the numbers, and

(c) Intensional definition is to provide a rule, that returns true or false depending

on whether the object is in the set or not.

[Readers may refer book on AI by Elaine Rich & Kevin Knight- page 122 - 123]

25

KR - issues • Finding Right structure

This is about access to right structure for describing a particular situation.

This requires, selecting an initial structure and then revising the choice.

While doing so, it is necessary to solve following problems :

− how to perform an initial selection of the most appropriate structure.

− how to fill in appropriate details from the current situations.

− how to find a better structure if the one chosen initially turns out not to

be appropriate.

− what to do if none of the available structures is appropriate.

− when to create and remember a new structure.

There is no good, general purpose method for solving all these problems.

Some knowledge representation techniques solve some of them.

[Readers may refer book on AI by Elaine Rich & Kevin Knight- page 124 - 126]

26

KR – using logic 2. KR Using Predicate Logic

In the previous section much has been illustrated about knowledge and KR

related issues. This section, illustrates how knowledge may be represented as

“symbol structures” that characterize bits of knowledge about objects, concepts,

facts, rules, strategies;

examples : “red” represents colour red;

“car1” represents my car ;

"red(car1)" represents fact that my car is red.

Assumptions about KR :

− Intelligent Behavior can be achieved by manipulation of symbol structures.

− KR languages are designed to facilitate operations over symbol structures,

have precise syntax and semantics;

Syntax tells which expression is legal ?,

e.g., red1(car1), red1 car1, car1(red1), red1(car1 & car2) ?; and

Semantic tells what an expression means ?

e.g., property “dark red” applies to my car.

− Make Inferences, draw new conclusions from existing facts.

To satisfy these assumptions about KR, we need formal notation that allow

automated inference and problem solving. One popular choice is use of logic.

27

KR – using Logic

• Logic

Logic is concerned with the truth of statements about the world.

Generally each statement is either TRUE or FALSE.

Logic includes : Syntax , Semantics and Inference Procedure.

◊ Syntax :

Specifies the symbols in the language about how they can be combined

to form sentences. The facts about the world are represented as

sentences in logic.

◊ Semantic :

Specifies how to assign a truth value to a sentence based on its

meaning in the world. It Specifies what facts a sentence refers to. A

fact is a claim about the world, and it may be TRUE or FALSE.

◊ Inference Procedure :

Specifies methods for computing new sentences from an existing

sentences.

Note :

Facts are claims about the world that are True or False.

Representation is an expression (sentence), stands for the objects and relations.

Sentences can be encoded in a computer program.

28

KR – using Logic

• Logic as a KR Language

Logic is a language for reasoning, a collection of rules used while doing

logical reasoning. Logic is studied as KR languages in artificial intelligence.

◊ Logic is a formal system in which the formulas or sentences have true

or false values.

◊ The problem of designing a KR language is a tradeoff between that

which is :

(a) Expressive enough to represent important objects and relations in

a problem domain.

(b) Efficient enough in reasoning and answering questions about

implicit information in a reasonable amount of time.

◊ Logics are of different types : Propositional logic, Predicate logic,

Temporal logic, Modal logic, Description logic etc;

They represent things and allow more or less efficient inference.

◊ Propositional logic and Predicate logic are fundamental to all logic.

Propositional Logic is the study of statements and their connectivity.

Predicate Logic is the study of individuals and their properties.

29

KR – Logic

2.1 Logic Representation

The Facts are claims about the world that are True or False.

Logic can be used to represent simple facts.

To build a Logic-based representation :

◊ User defines a set of primitive symbols and the associated semantics.

◊ Logic defines ways of putting symbols together so that user can define

legal sentences in the language that represent TRUE facts.

◊ Logic defines ways of inferring new sentences from existing ones.

◊ Sentences - either TRUE or false but not both are called propositions.

◊ A declarative sentence expresses a statement with a proposition as

content; example:

the declarative "snow is white" expresses that snow is white;

further, "snow is white" expresses that snow is white is TRUE.

In this section, first Propositional Logic (PL) is briefly explained and then

the Predicate logic is illustrated in detail.

30

KR - Propositional Logic • Propositional Logic (PL)

A proposition is a statement, which in English would be a declarative

sentence. Every proposition is either TRUE or FALSE.

Examples: (a) The sky is blue., (b) Snow is cold. , (c) 12 * 12=144

‡ propositions are “sentences” , either true or false but not both.

‡ a sentence is smallest unit in propositional logic.

‡ if proposition is true, then truth value is "true" .

if proposition is false, then truth value is "false" .

Example : Sentence Truth value Proposition (Y/N)

"Grass is green" "true" Yes

"2 + 5 = 5" "false" Yes

"Close the door" - No

"Is it hot out side ?" - No

"x > 2" where is variable - No (since x is not defined) "x = x" - No

(don't know what is "x" and "="; "3 = 3" or "air is equal to air" or

"Water is equal to water" has no meaning)

− Propositional logic is fundamental to all logic.

− Propositional logic is also called Propositional calculus, Sentential

calculus, or Boolean algebra.

− Propositional logic tells the ways of joining and/or modifying entire

propositions, statements or sentences to form more complicated

propositions, statements or sentences, as well as the logical

relationships and properties that are derived from the methods of

combining or altering statements.

31

KR - Propositional Logic

■ Statement, variables and symbols

These and few more related terms, such as, connective, truth value,

contingencies, tautologies, contradictions, antecedent, consequent and

argument are explained below.

◊ Statement

Simple statements (sentences), TRUE or FALSE, that does not

contain any other statement as a part, are basic propositions;

lower-case letters, p, q, r, are symbols for simple statements.

Large, compound or complex statement are constructed from basic

propositions by combining them with connectives.

◊ Connective or Operator

The connectives join simple statements into compounds, and joins

compounds into larger compounds.

Table below indicates, five basic connectives and their symbols :

− listed in decreasing order of operation priority;

− operations with higher priority is solved first.

Example of a formula : ((((a Λ ¬b) V c → d) ↔ ¬ (a V c ))

Connectives and Symbols in decreasing order of operation priority

Connective Symbols Read as

assertion P "p is true"

negation ¬p ~ ! NOT "p is false"

conjunction p ∧ q · && & AND "both p and q are true"

disjunction P v q || | OR "either p is true, or q is true, or both "

implication p → q ⊃ ⇒ if ..then "if p is true, then q is true" " p implies q "

equivalence ↔ ≡ ⇔ if and only if "p and q are either both true or both false"

Note : The propositions and connectives are the basic elements of

propositional logic.

32

KR - Propositional Logic

◊ Truth value The truth value of a statement is its TRUTH or FALSITY ,

Example :

p is either TRUE or FALSE,

~p is either TRUE or FALSE,

p v q is either TRUE or FALSE, and so on.

use " T " or " 1 " to mean TRUE.

use " F " or " 0 " to mean FALSE Truth table defining the basic connectives :

p q ¬ p ¬ q p ∧ q p v q p→q p ↔ q q→p

T T F F T T T T T

T F F T F T F F T

F T T F F T T F F

F F T T F F T T T

33

KR - Propositional Logic ◊ Tautologies

A proposition that is always true is called a tautology.

e.g., (P v ¬P) is always true regardless of the truth value of the

proposition P.

◊ Contradictions

A proposition that is always false is called a contradiction.

e.g., (P ∧ ¬P) is always false regardless of the truth value of

the proposition P.

◊ Contingencies

A proposition is called a contingency, if that proposition is neither

a tautology nor a contradiction

e.g., (P v Q) is a contingency.

◊ Antecedent, Consequent

In the conditional statements, p → q , the

1st statement or "if - clause" (here p) is called antecedent ,

2nd statement or "then - clause" (here q) is called consequent.

34

KR - Propositional Logic

◊ Argument

Any argument can be expressed as a compound statement.

Take all the premises, conjoin them, and make that conjunction the

antecedent of a conditional and make the conclusion the

consequent. This implication statement is called the corresponding

conditional of the argument.

Note :

− Every argument has a corresponding conditional, and every

implication statement has a corresponding argument.

− Because the corresponding conditional of an argument is a statement,

it is therefore either a tautology, or a contradiction, or a contingency.

‡ An argument is valid "if and only if" its corresponding conditional

is a tautology.

‡ Two statements are consistent "if and only if" their conjunction

is not a contradiction.

‡ Two statements are logically equivalent "if and only if" their

truth table columns are identical; "if and only if" the statement

of their equivalence using " ≡ " is a tautology.

Note : The truth tables are adequate to test validity, tautology,

contradiction, contingency, consistency, and equivalence.

35

KR - Predicate Logic • Predicate logic

The propositional logic, is not powerful enough for all types of assertions;

Example : The assertion "x > 1", where x is a variable, is not a proposition

because it is neither true nor false unless value of x is defined.

For x > 1 to be a proposition ,

− either we substitute a specific number for x ;

− or change it to something like

"There is a number x for which x > 1 holds";

− or "For every number x, x > 1 holds".

Consider example :

“All men are mortal.

Socrates is a man.

Then Socrates is mortal” ,

These cannot be expressed in propositional logic as a finite and logically

valid argument (formula).

We need languages : that allow us to describe properties (predicates) of

objects, or a relationship among objects represented by the variables .

Predicate logic satisfies the requirements of a language.

− Predicate logic is powerful enough for expression and reasoning.

− Predicate logic is built upon the ideas of propositional logic.

36

KR - Predicate Logic ■ Predicate :

Every complete sentence contains two parts: a subject and a predicate.

The subject is what (or whom) the sentence is about.

The predicate tells something about the subject;

Example :

A sentence "Judy {runs}".

The subject is Judy and the predicate is runs .

Predicate, always includes verb, tells something about the subject.

Predicate is a verb phrase template that describes a property of objects,

or a relation among objects represented by the variables.

Example:

“The car Tom is driving is blue" ;

"The sky is blue" ;

"The cover of this book is blue"

Predicate is “is blue" , describes property.

Predicates are given names; Let ‘B’ is name for predicate "is_blue".

Sentence is represented as "B(x)" , read as "x is blue";

“x” represents an arbitrary Object .

37

KR - Predicate Logic

■ Predicate logic expressions :

The propositional operators combine predicates, like

If ( p(....) && ( !q(....) || r (....) ) )

Examples of logic operators : disjunction (OR) and conjunction (AND).

Consider the expression with the respective logic symbols || and &&

x < y || ( y < z && z < x)

Which is true || ( true && true) ;

Applying truth table, found True Assignment for < are 3, 2, 1 for x, y, z and then

the value can be FALSE or TRUE

3 < 2 || ( 2 < 1 && 1 < 3)

It is False

38

KR - Predicate Logic

■ Predicate Logic Quantifiers

As said before, x > 1 is not proposition and why ?

Also said, that for x > 1 to be a proposition what is required ?

Generally, a predicate with variables (is called atomic formula) can be

made a proposition by applying one of the following two operations to

each of its variables :

1. Assign a value to the variable; e.g., x > 1, if 3 is assigned to x

becomes 3 > 1 , and it then becomes a true statement, hence a

proposition.

2. Quantify the variable using a quantifier on formulas of predicate logic

(called wff ), such as x > 1 or P(x), by using Quantifiers on variables.

Apply Quantifiers on Variables

‡ Variable x

* x > 5 is not a proposition, its truth depends upon the

value of variable x * to reason such statements, x need to be declared

‡ Declaration x : a * x : a declares variable x * x : a read as “x is an element of set a”

‡ Statement p is a statement about x * Q x : a • p is quantification of statement

statement

declaration of variable x as element of set a

quantifier

* Quantifiers are two types :

universal quantifiers , denoted by symbol and

existential quantifiers , denoted by symbol

Note : The next few slide tells more on these two Quantifiers.

39

KR - Predicate Logic

■ Universe of Discourse

The universe of discourse, also called domain of discourse or universe.

This indicates :

− a set of entities that the quantifiers deal.

− entities can be set of real numbers, set of integers, set of all cars on

a parking lot, the set of all students in a classroom etc.

− universe is thus the domain of the (individual) variables.

− propositions in the predicate logic are statements on objects of a

universe.

The universe is often left implicit in practice, but it should be obvious

from the context.

Examples:

− About natural numbers forAll x, y (x < y or x = y or x > y),

there is no need to be more precise and say forAll x, y in N,

because N is implicit, being the universe of discourse.

− About a property that holds for natural numbers but not for real

numbers, it is necessary to qualify what the allowable values of x

and y are.

40

KR - Predicate Logic

■ Apply Universal quantifier " For All "

Universal Quantification allows us to make a statement about a

collection of objects.

‡ Universal quantification: x : a • p * read “ for all x in a , p holds ” * a is universe of discourse

* x is a member of the domain of discourse.

* p is a statement about x ‡ In propositional form it is written as : x P(x)

* read “ for all x, P(x) holds ”

“ for each x, P(x) holds ” or

“ for every x, P(x) holds ” * where P(x) is predicate,

x means all the objects x in the universe

P(x) is true for every object x in the universe

‡ Example : English language to Propositional form

* "All cars have wheels" x : car • x has wheel

* x P(x) where P (x) is predicate tells : ‘x has wheels’

x is variable for object ‘cars’ that populate universe of discourse

41

KR - Predicate Logic ■ Apply Existential quantifier " There Exists "

Existential Quantification allows us to state that an object does exist

without naming it.

‡ Existential quantification: x : a • p

* read “ there exists an x such that p holds ” * a is universe of discourse * x is a member of the domain of discourse. * p is a statement about x ‡ In propositional form it is written as : x P(x)

* read “ there exists an x such that P(x) ” or

“ there exists at least one x such that P(x) ” * Where P(x) is predicate

x means at least one object x in the universe

P(x) is true for least one object x in the universe ‡ Example : English language to Propositional form

* “ Someone loves you ”

x : Someone • x loves you

* x P(x) where P(x) is predicate tells : ‘ x loves you ’

x is variable for object ‘ someone ’ that populate universe of discourse

42

KR - Predicate Logic ■ Formula :

In mathematical logic, a formula is a type of abstract object, a token of

which is a symbol or string of symbols which may be interpreted as any

meaningful unit in a formal language.

‡ Terms :

Defined recursively as variables, or constants, or functions like

f(t1, . . . , tn), where f is an n-ary function symbol, and t1, . . . , tn

are terms. Applying predicates to terms produce atomic formulas.

‡ Atomic formulas :

An atomic formula (or simply atom) is a formula with no deeper

propositional structure, i.e., a formula that contains no logical

connectives or a formula that has no strict sub-formulas.

− Atoms are thus the simplest well-formed formulas of the logic.

− Compound formulas are formed by combining the atomic formulas

using the logical connectives.

− Well-formed formula ("wiff") is a symbol or string of symbols (a

formula) generated by the formal grammar of a formal language.

An atomic formula is one of the form:

− t1 = t2, where t1 and t2 are terms, or

− R(t1, . . . , tn), where R is an n-ary relation symbol, and t1, . . . , tn

are terms.

− ¬ a is a formula when a is a formula.

− (a ∧ b) and (a v b) are formula when a and b are formula

‡ Compound formula : example

((((a ∧ b ) ∧ c) ∨ ((¬ a ∧ b) ∧ c)) ∨ ((a ∧ ¬ b) ∧ c))

43

KR – logic relation

2.2 Representing “ IsA ” and “ Instance ” Relationships

Logic statements, containing subject, predicate, and object, were explained.

Also stated, two important attributes "instance" and "isa", in a hierarchical

structure (ref. fig. Inheritable KR). These two attributes support property

inheritance and play important role in knowledge representation.

The ways, attributes "instance" and "isa", are logically expressed are :

■ Example : A simple sentence like "Joe is a musician" ◊ Here "is a" (called IsA) is a way of expressing what logically is called

a class-instance relationship between the subjects represented by

the terms "Joe" and "musician".

◊ "Joe" is an instance of the class of things called "musician".

"Joe" plays the role of instance,

"musician" plays the role of class in that sentence.

◊ Note : In such a sentence, while for a human there is no confusion,

but for computers each relationship have to be defined explicitly.

This is specified as: [Joe] IsA [Musician]

i.e., [Instance] IsA [Class] 44

KR – functions & predicates 2.3 Computable Functions and Predicates

The objective is to define class of functions C computable in terms of F.

This is expressed as C { F } is explained below using two examples :

(1) "evaluate factorial n" and (2) "expression for triangular functions".

■ Example (1) : A conditional expression to define factorial n ie n! ◊ Expression

“ if p1 then e1 else if p2 then e2 . . . else if pn then en” .

ie. (p1 → e1, p2 → e2, . . . . . . pn → en )

Here p1, p2, . . . . pn are propositional expressions taking the

values T or F for true and false respectively. ◊ The value of ( p1 → e1, p2 → e2, . . . . . .pn → en ) is the value of

the e corresponding to the first p that has value T. ◊ The expressions defining n! , n= 5, recursively are :

n! = n x (n-1)! for n ≥ 1 5! = 1 x 2 x 3 x 4 x 5 = 120 0! = 1

The above definition incorporates an instance that the product of

no numbers ie 0! = 1 , then only, the recursive relation (n + 1)! =

n! x (n+1) works for n = 0 . ◊ Now use conditional expressions

n! = ( n = 0 → 1, n ≠ 0 → n . (n – 1 ) ! )

to define functions recursively. ◊ Example: Evaluate 2! according to above definition. 2! = ( 2 = 0 → 1, 2 ≠ 0 → 2 . ( 2 – 1 )! )

= 2 x 1! = 2 x ( 1 = 0 → 1, 1 ≠ 0 → 1 . ( 1 – 1 )! )

= 2 x 1 x 0! = 2 x 1 x ( 0 = 0 → 1, 0 ≠ 0 → 0 . ( 0 – 1 )! )

= 2 x 1 x 1 = 2 45

KR – functions & predicates ■ Example (2) : A conditional expression for triangular functions

◊ The graph of a well known triangular function is shown below

the conditional expressions for triangular functions are

x = (x < 0 → -x , x ≥ 0 → x)

◊ the triangular function of the above graph is represented by the

conditional expression is

tri (x) = (x ≤ -1 → 0, x ≤ 0 → -x, x ≤ 1 → x, x > 1 → 0)

46

Y

-1,0 1,0X

0,1

KR - Predicate Logic – resolution 2.4 Resolution

Resolution is a procedure used in proving that arguments which are

expressible in predicate logic are correct.

Resolution is a procedure that produces proofs by refutation or

contradiction.

Resolution lead to refute a theorem-proving technique for sentences in

propositional logic and first-order logic.

− Resolution is a rule of inference.

− Resolution is a computerized theorem prover.

− Resolution is so far only defined for Propositional Logic. The strategy is

that the Resolution techniques of Propositional logic be adopted in

Predicate Logic.

47

KR Using Rules 3. KR Using Rules

Knowledge representations using predicate logic have been illustrated.

The other most popular approach to Knowledge representation is to use

production rules, sometimes called IF-THEN rules. The remaining two other

types of KR are semantic net and frames.

The production rules are simple but powerful forms of knowledge representation

providing the flexibility of combining declarative and procedural representation

for using them in a unified form.

Examples of production rules :

− IF condition THEN action

− IF premise THEN conclusion

− IF proposition p1 and proposition p2 are true THEN proposition p3 is true

The advantages of production rules :

− they are modular,

− each rule define a small and independent piece of knowledge.

− new rules may be added and old ones deleted

− rules are usually independently of other rules.

The production rules as knowledge representation mechanism are used in the

design of many "Rule-based systems" also called "Production systems" .

48

KR Using Rules

• Types of rules

Three major types of rules used in the Rule-based production systems.

■ Knowledge Declarative Rules :

These rules state all the facts and relationships about a problem.

e.g., IF inflation rate declines

THEN the price of gold goes down.

These rules are a part of the knowledge base.

■ Inference Procedural Rules

These rules advise on how to solve a problem, while certain facts are

known.

e.g., IF the data needed is not in the system

THEN request it from the user.

These rules are part of the inference engine.

■ Meta rules

These are rules for making rules. Meta-rules reason about which rules

should be considered for firing.

e.g., IF the rules which do not mention the current goal in their premise,

AND there are rules which do mention the current goal in their premise,

THEN the former rule should be used in preference to the latter.

− Meta-rules direct reasoning rather than actually performing

reasoning.

− Meta-rules specify which rules should be considered and in which

order they should be invoked.

49

KR – procedural & declarative 3.1 Procedural versus Declarative Knowledge

These two types of knowledge were defined in earlier slides.

■ Procedural Knowledge : knowing 'how to do'

Includes : Rules, strategies, agendas, procedures, models.

These explains what to do in order to reach a certain conclusion.

e.g., Rule: To determine if Peter or Robert is older, first find their ages.

It is knowledge about how to do something. It manifests itself in the

doing of something, e.g., manual or mental skills cannot reduce to

words. It is held by individuals in a way which does not allow it to be

communicated directly to other individuals.

Accepts a description of the steps of a task or procedure. It Looks

similar to declarative knowledge, except that tasks or methods are

being described instead of facts or things.

■ Declarative Knowledge : knowing 'what', knowing 'that'

Includes : Concepts, objects, facts, propositions, assertions, models.

It is knowledge about facts and relationships, that

− can be expressed in simple and clear statements,

− can be added and modified without difficulty.

e.g., A car has four tyres; Peter is older than Robert.

Declarative knowledge and explicit knowledge are articulated knowledge

and may be treated as synonyms for most practical purposes.

Declarative knowledge is represented in a format that can be

manipulated, decomposed and analyzed independent of its content.

50

KR – procedural & declarative ■ Comparison :

Comparison between Procedural and Declarative Knowledge :

Procedural Knowledge Declarative Knowledge

• Hard to debug • Easy to validate

• Black box • White box

• Obscure • Explicit

• Process oriented • Data - oriented

• Extension may effect stability • Extension is easy

• Fast , direct execution • Slow (requires interpretation)

• Simple data type can be used • May require high level data type

• Representations in the form of

sets of rules, organized into

routines and subroutines.

• Representations in the form of

production system, the entire set

of rules for executing the task.

51

KR – procedural & declarative ■ Comparison :

Comparison between Procedural and Declarative Language :

Procedural Language Declarative Language

• Basic, C++, Cobol, etc. • SQL

• Most work is done by interpreter of the languages

• Most work done by Data Engine within the DBMS

• For one task many lines of code

• For one task one SQL statement

• Programmer must be skilled in translating the objective into lines of procedural code

• Programmer must be skilled in clearly stating the objective as a SQL statement

• Requires minimum of management around the actual data

• Relies on SQL-enabled DBMS to hold the data and execute the SQL statement .

• Programmer understands and has access to each step of the code

• Programmer has no interaction with the execution of the SQL statement

• Data exposed to programmer during execution of the code

• Programmer receives data at end as an entire set

• More susceptible to failure due to changes in the data structure

• More resistant to changes in the data structure

• Traditionally faster, but that is changing

• Originally slower, but now setting speed records

• Code of procedure tightly linked to front end

• Same SQL statements will work with most front endsCode loosely linked to front end.

• Code tightly integrated with structure of the data store

• Code loosely linked to structure of data; DBMS handles structural issues

• Programmer works with a pointer or cursor

• Programmer not concerned with positioning

• Knowledge of coding tricks applies only to one language

• Knowledge of SQL tricks applies to any language using SQL

52

KR – Logic Programming 3.2 Logic Programming

Logic programming offers a formalism for specifying a computation in terms

of logical relations between entities.

− logic program is a collection of logic statements.

− programmer describes all relevant logical relationships between the

various entities.

− computation determines whether or not, a particular conclusion follows

from those logical statements.

• characteristics of Logic program

Logic program is characterized by set of relations and inferences.

− the program consists of a set of axioms and a goal statement.

− the Rules of inference determine whether the axioms are sufficient to

ensure the truth of the goal statement.

− the execution of a logic program corresponds to the construction of a

proof of the goal statement from the axioms.

− the Programmer specify basic logical relationships, does not specify the

manner in which inference rules are applied.

Thus Logic + Control = Algorithms

• Examples of Logic Statements

− Statement

A grand-parent is a parent of a parent.

− Statement expressed in more closely related logic terms as

A person is a grand-parent if she/he has a child and

that child is a parent.

− Statement expressed in first order logic as

(for all) x: grand-parent(x) ← (there exist) y, z : parent(x, y) &

parent(y, z) 53

KR – Logic Programming • Logic programming Language

A programming language includes :

− the syntax

− the semantics of programs and

− the computational model.

There are many ways of organizing computations. The most familiar paradigm is procedural. The program specifies a

computation by saying "how" it is to be performed. FORTRAN, C, and

object-oriented languages fall under this general approach.

Another paradigm is declarative. The program specifies a computation by

giving the properties of a correct answer. Prolog and logic data language

(LDL) are examples of declarative languages, emphasize the logical

properties of a computation.

Prolog and LDL are called logic programming languages.

PROLOG is the most popular Logic programming system.

54

KR – Logic Programming • Syntax and terminology (relevant to Prolog programs)

In any language, the formation of components (expressions, statements,

etc.), is guided by syntactic rules. The components are divided into two

parts: (A) data components and (B) program components.

(A) Data components :

Data components are collection of data objects that follow hierarchy.

Data object of any kind is also called

a term. A term is a constant, a variable

or a compound term.

Simple data object is not

decomposable; e.g. atoms, numbers,

constants, variables. The syntax

distinguishes the data objects, hence

no need for declaring them.

Structured data object are made of

several components; e.g. general,

special structure.

All these data components were mentioned in the earlier slides, are

now explained in detail below. 55

Data Objects(terms)

Simple Structured

VariablesConstants

NumbersAtoms

KR – Logic Programming (a) Data objects : The data objects of any kind is called a term.

◊ Term : examples

‡ Constants: denote elements such as integers, floating point, atoms.

‡ Variables: denote a single but unspecified element; symbols for

variables begin with an uppercase letter or an underscore.

‡ Compound terms: comprise a functor and sequence of one or more

compound terms called arguments.

► Functor : is characterized by its name, which is an atom, and its

arity or number of arguments.

ƒ/n = ƒ( t1 , t2, . . . tn )

where ƒ is name of the functor and is of arity n

ti 's are the arguments

ƒ/n denotes functor ƒ of arity n

Functors with the same name but different arities are distinct.

‡ Ground and non-ground: Terms are ground if they contain no

variables; otherwise they are non-ground. Goals are atoms or

compound terms, and are generally non-ground.

56

KR – Logic Programming (b) Simple data objects : Atoms, Numbers, Variables

◊ Atoms

‡ a lower-case letter, possibly followed by other letters (either case),

digits, and underscore character.

e.g. a greaterThan two_B_or_not_2_b

‡ a string of special characters such as: + - * / \ = ^ < > : . ~ @ # $ &

e.g. <> ##&& ::=

‡ a string of any characters enclosed within single quotes.

e.g. 'ABC' '1234' 'a<>b'

‡ following are also atoms ! ; [] {}

◊ Numbers

‡ applications involving heavy numerical calculations are rarely

written in Prolog.

‡ integer representation: e.g. 0 -16 33 +100

‡ real numbers written in standard or scientific notation,

e.g. 0.5 -3.1416 6.23e+23 11.0e-3 -2.6e-2

◊ Variables

‡ begins by a capital letter, possibly followed by other letters (either

case), digits, and underscore character.

e.g. X25 List Noun_Phrase

57

KR – Logic Programming (c) Structured data objects : General Structures , Special Structures

◊ General Structures

‡ a structured term is syntactically formed by a functor and a list of

arguments.

‡ functor is an atom.

‡ list of arguments appears between parentheses.

‡ arguments are separated by a comma.

‡ each argument is a term (i.e., any Prolog data object).

‡ the number of arguments of a structured term is called its arity.

‡ e.g. greaterThan(9, 6) f(a, g(b, c), h(d)) plus(2, 3, 5)

Note : a structure in Prolog is a mechanism for combining terms together,

like integers 2, 3, 5 are combined with the functor plus.

◊ Special Structures

‡ In Prolog an ordered collection of terms is called a list .

‡ Lists are structured terms and Prolog offers a convenient notation to

represent them:

* Empty list is denoted by the atom [ ].

* Non-empty list carries element(s) between square brackets,

separating elements by comma.

e.g. [bach, bee] [apples, oranges, grapes] []

58

KR – Logic Programming (B) Program Components

A Prolog program is a collection of predicates or rules. A predicate

establishes a relationships between objects.

(a)

Clause, Predicate, Sentence, Subject

‡ Clause is a collection of grammatically-related words .

‡ Predicate is composed of one or more clauses.

‡ Clauses are the building blocks of sentences; every sentence contains

one or more clauses.

‡ A Complete Sentence has two parts: subject and predicate.

o subject is what (or whom) the sentence is about.

o predicate tells something about the subject.

‡ Example 1 : "cows eat grass".

It is a clause, because it contains the subject "cows" and

the predicate "eat grass."

‡ Example 2 : "cows eating grass are visible from highway"

This is a complete clause. The subject "cows eating grass" and

the predicate "are visible from the highway" makes complete thought. 59

KR – Logic Programming (b)

Predicates & Clause

Syntactically a predicate is composed of one or more clauses.

‡ The general form of clauses is :

<left-hand-side> :- <right-hand-side>.

where LHS is a single goal called "goal" and

RHS is composed of one or more goals, separated by commas, called "sub-goals" of the goal on left-hand side.

‡ The structure of a clause in logic program head body

pred ( functor(var1, var2)) :- pred(var1) , pred(var2)

literal literal

clause

‡ Example : grand_parent (X, Y) :- parent(X, Z), parent(Z, Y).

parent (X, Y) :- mother(X, Y).

parent (X, Y) :- father(X, Y).

‡ Interpretation:

* a clause specifies the conditional truth of the goal on the LHS;

i.e., goal on LHS is assumed to be true if the sub-goals on RHS are all

true. A predicate is true if at least one of its clauses is true.

* An individual "Y" is the grand-parent of "X" if a parent of that same

"X" is "Z" and "Y" is the parent of that "Z".

Y Z X

* An individual "Y" is a parent of "X" if "Y" is the mother of "X"

Y X

* An individual "Y" is a parent of "X" if "Y" is the father of "X".

Y X

60

(Z is parent of X) (Y is parent of Z)

(Y is grand parent of X)

(Y is parent of X)

(Y is mother of X)

(Y is parent of X)

(Y is father of X)

KR – Logic Programming (c)

Unit Clause - a special Case

Unlike the previous example of conditional truth, one often encounters

unconditional relationships that hold.

‡ In Prolog the clauses that are unconditionally true are called unit

clause or fact

‡ Example : Unconditionally relationships

say 'Y' is the father of 'X' is unconditionally true.

This relationship as a Prolog clause is :

father(X, Y) :- true.

Interpreted as relationship of father between Y and X is always true;

or simply stated as Y is father of X

‡ Goal true is built-in in Prolog and always holds.

‡ Prolog offers a simpler syntax to express unit clause or fact

father(X, Y)

ie the :- true part is simply omitted.

61

KR – Logic Programming (d)

Queries

In Prolog the queries are statements called directive. A special case of

directives, are called queries.

‡ Syntactically, directives are clauses with an empty left-hand side.

Example : ? - grandparent(X, W).

This query is interpreted as : Who is a grandparent of X ?

By issuing queries, Prolog tries to establish the validity of specific

relationships.

‡ The result of executing a query is either success or failure

Success, means the goals specified in the query holds according to the

facts and rules of the program.

Failure, means the goals specified in the query does not hold according

to the facts and rules of the program.

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KR – Logic - models of computation • Programming paradigms : Models of Computation

A complete description of a programming language includes the

computational model, syntax, semantics, and pragmatic considerations that

shape the language.

Models of Computation :

A computational model is a collection of values and operations, while

computation is the application of a sequence of operations to a value

to yield another value. There are three basic computational models :

(a) Imperative, (b) Functional, and (c) Logic. In addition to these,

there are two programming paradigms (concurrent and object-oriented

programming). While, they are not models of computation, they rank in

importance with computational models.

63

KR – Logic - models of computation (a) Imperative Model :

The Imperative model of computation, consists of a state and an

operation of assignment which is used to modify the state. Programs

consist of sequences of commands. The computations are changes in

the state.

Example 1 : Linear function

A linear function y = 2x + 3 can be written as

Y := 2 ∗ X + 3

The implementation determines the value of X in the state and then

create a new state, which differs from the old state. The value of Y in

the new state is the value that 2 ∗ X + 3 had in the old state.

Old State: X = 3, Y = -2,

Y := 2 ∗ X + 3

New State: X = 3, Y = 9, The imperative model is closest to the hardware model on which

programs are executed, that makes it most efficient model in terms

of execution time. 64

KR – Logic - models of computation (b) Functional model :

The Functional model of computation, consists of a set of values,

functions, and the operation of functions. The functions may be named

and may be composed with other functions. They can take other

functions as arguments and return results. The programs consist of

definitions of functions. The computations are application of functions to

values.

‡ Example 1 : Linear function

A linear function y = 2x + 3 can be defined as :

f (x) = 2 ∗ x + 3

‡ Example 2 : Determine a value for Circumference.

Assigned a value to Radius, that determines a value for

Circumference.

Circumference = 2 × pi × radius where pi = 3.14

Generalize Circumference with the variable "radius" ie

Circumference(radius) = 2 × pi × radius , where pi = 3.14

Functional models are developed over many years. The notations and

methods form the base upon which problem solving methodologies rest. 65

KR – Logic - models of computation (c) Logic model :

The logic model of computation is based on relations and logical

inference. Programs consist of definitions of relations. Computations are

inferences (is a proof).

‡ Example 1 : Linear function

A linear function y = 2x + 3 can be represented as :

f (X , Y) if Y is 2 ∗ X + 3.

‡ Example 2: Determine a value for Circumference.

The earlier circumference computation can be represented as:

Circle (R , C) if Pi = 3.14 and C = 2 ∗ pi ∗ R.

The function is represented as a relation between radious R and

circumference C. ‡ Example 3: Determine the mortality of Socrates.

The program is to determine the mortality of Socrates.

The fact given that Socrates is human.

The rule is that all humans are mortal, that is

for all X, if X is human then X is mortal.

To determine the mortality of Socrates, make the assumption that

there are no mortals, that is ¬ mortal (Y)

[logic model continued in the next slide] 66

KR – Logic - models of computation [logic model continued in the previous slide] ‡ The fact and rule are:

human (Socrates)

mortal (X) if human (X)

‡ To determine the mortality of Socrates, make the assumption that

there are no mortals i.e. ¬ mortal (Y) ‡ Computation (proof) that Socrates is mortal :

1. human(Socrates) Fact

2. mortal(X) if human(X) Rule

3 ¬mortal(Y) assumption

4.(a) X = Y from 2 & 3 by unification

4.(b) ¬human(Y) and modus tollens

5. Y = Socrates from 1 and 4 by unification

6. Contradiction 5, 4b, and 1

‡ Explanation : * The 1st line is the statement "Socrates is a man."

* The 2nd line is a phrase "all human are mortal"

into the equivalent "for all X, if X is a man then X is mortal".

* The 3rd line is added to the set to determine the mortality of Socrates.

* The 4th line is the deduction from lines 2 and 3. It is justified by the

inference rule modus tollens which states that if the conclusion of a rule is known to be false, then so is the hypothesis.

* Variables X and Y are unified because they have same value.

* By unification, Lines 5, 4b, and 1 produce contradictions and identify Socrates as mortal.

* Note that, resolution is the an inference rule which looks for a contradiction and it is facilitated by unification which determines if there is a substitution which makes two terms the same.

Logic model formalizes the reasoning process. It is related to relational

data bases and expert systems. 67

KR – forward-backward reasoning 3.3 Forward versus Backward Reasoning

Rule-Based system architecture consists a set of rules, a set of facts, and

an inference engine. The need is to find what new facts can be derived.

Given a set of rules, there are essentially two ways to generate new

knowledge: one, forward chaining and the other, backward chaining.

■ Forward chaining : also called data driven.

It starts with the facts, and sees what rules apply.

■ Backward chaining : also called goal driven.

It starts with something to find out, and looks for rules that will help in

answering it. 68

KR – forward-backward reasoning ■ Example 1 :

Rule R1 : IF hot AND smoky THEN fire

Rule R2 : IF alarm_beeps THEN smoky

Rule R3 : IF fire THEN switch_on_sprinklers

Fact F1 : alarm_beeps [Given]

Fact F2 : hot [Given]

■ Example 2 :

Rule R1 : IF hot AND smoky THEN ADD fire

Rule R2 : IF alarm_beeps THEN ADD smoky

Rule R3 : IF fire THEN ADD switch_on_sprinklers

Fact F1 : alarm_beeps [Given]

Fact F2 : hot [Given]

69

KR – forward-backward reasoning ■ Example 3 : A typical Forward Chaining

Rule R1 : IF hot AND smoky THEN ADD fire

Rule R2 : IF alarm_beeps THEN ADD smoky

Rule R3 : If fire THEN ADD switch_on_sprinklers

Fact F1 : alarm_beeps [Given]

Fact F2 : hot [Given]

Fact F4 : smoky [from F1 by R2]

Fact F2 : fire [from F2, F4 by R1]

Fact F6 : switch_on_sprinklers [from F4 by R3]

■ Example 4 : A typical Backward Chaining

Rule R1 : IF hot AND smoky THEN fire

Rule R2 : IF alarm_beeps THEN smoky

Rule R3 : If _re THEN switch_on_sprinklers

Fact F1 : hot [Given]

Fact F2 : alarm_beeps [Given]

Goal : Should I switch sprinklers on?

70

KR – forward chaining • Forward chaining

The Forward chaining system, properties , algorithms, and conflict

resolution strategy are illustrated.

■ Forward chaining system

‡ facts are held in a working memory ‡ condition-action rules represent actions to be taken when specified

facts occur in working memory. ‡ typically, actions involve adding or deleting facts from the working

memory.

■ Properties of Forward Chaining

‡ all rules which can fire do fire. ‡ can be inefficient - lead to spurious rules firing, unfocused problem

solving

‡ set of rules that can fire known as conflict set. ‡ decision about which rule to fire is conflict resolution. 71

rules

Working Memory

Inference Engine

RuleBase

User

facts

facts

facts

KR – forward chaining ■ Forward chaining algorithm - I

Repeat

‡ Collect the rule whose condition matches a fact in WM.

‡ Do actions indicated by the rule.

(add facts to WM or delete facts from WM)

Until problem is solved or no condition match

Apply on the Example 2 extended (adding 2 more rules and 1 fact)

Rule R1 : IF hot AND smoky THEN ADD fire

Rule R2 : IF alarm_beeps THEN ADD smoky

Rule R3 : If fire THEN ADD switch_on_sprinklers

Rule R4 : IF dry THEN ADD switch_on_humidifier

Rule R5 : IF sprinklers_on THEN DELETE dry

Fact F1 : alarm_beeps [Given]

Fact F2 : hot [Given]

Fact F2 : Dry [Given]

Now, two rules can fire (R2 and R4)

‡ R4 fires, humidifier is on (then, as before)

‡ R2 fires, humidifier is off A conflict !

■ Forward chaining algorithm - II Repeat

‡ Collect the rules whose conditions match facts in WM.

‡ If more than one rule matches

◊ Use conflict resolution strategy to eliminate all but one

‡ Do actions indicated by the rules

(add facts to WM or delete facts from WM)

Until problem is solved or no condition match

72

KR – forward chaining ■ Conflict Resolution Strategy

Conflict set is the set of rules that have their conditions satisfied by

working memory elements. Conflict resolution normally selects a

single rule to fire. The popular conflict resolution mechanisms are

Refractory, Recency, Specificity.

◊ Refractory

‡ a rule should not be allowed to fire more than once on the same data.

‡ discard executed rules from the conflict set.

‡ prevents undesired loops.

◊ Recency

‡ rank instantiations in terms of the recency of the elements in the

premise of the rule.

‡ rules which use more recent data are preferred.

‡ working memory elements are time-tagged indicating at what cycle

each fact was added to working memory.

◊ Specificity

‡ rules which have a greater number of conditions and are therefore

more difficult to satisfy, are preferred to more general rules with fewer

conditions.

‡ more specific rules are ‘better’ because they take more of the data

into account.

73

KR – forward chaining ■ Alternative to Conflict Resolution – Use Meta Knowledge

Instead of conflict resolution strategies, sometimes we want to use

knowledge in deciding which rules to fire. Meta-rules reason about

which rules should be considered for firing. They direct reasoning rather

than actually performing reasoning. ‡ Meta-knowledge : knowledge about knowledge to guide search.

‡ Example of meta-knowledge

IF conflict set contains any rule (c , a) such that

a = "animal is mammal''

THEN fire (c , a)

‡ This example says meta-knowledge encodes knowledge about how

to guide search for solution. ‡ Meta-knowledge, explicitly coded in the form of rules with "object

level" knowledge. 74

KR – backward chaining • Backward chaining

Backward chaining system and the algorithm are illustrated.

■ Backward chaining system

‡ Backward chaining means reasoning from goals back to facts.

The idea is to focus on the search. ‡ Rules and facts are processed using backward chaining interpreter. ‡ Checks hypothesis, e.g. "should I switch the sprinklers on?"

■ Backward chaining algorithm

‡ Prove goal G : If G is in the initial facts , it is proven. Otherwise, find a rule which can be used to conclude G, and

try to prove each of that rule's conditions.

Encoding of rules

Rule R1 : IF hot AND smoky THEN fire

Rule R2 : IF alarm_beeps THEN smoky

Rule R3 : If fire THEN switch_on_sprinklers

Fact F1 : hot [Given]

Fact F2 : alarm_beeps [Given]

Goal : Should I switch sprinklers on?

75

alarm_beeps

Smoky hot

switch_on_sprinklers

fire

KR – backward chaining • Forward vs Backward Chaining

‡ Depends on problem, and on properties of rule set.

‡ If there is clear hypotheses, then backward chaining is likely to be

better; e.g., Diagnostic problems or classification problems, Medical

expert systems

‡ Forward chaining may be better if there is less clear hypothesis and

want to see what can be concluded from current situation; e.g.,

Synthesis systems - design / configuration. 76

KR – control knowledge 3.4 Control Knowledge

An algorithm consists of : logic component, that specifies the knowledge

to be used in solving problems, and control component, that determines

the problem-solving strategies by means of which that knowledge is used.

Thus Algorithm = Logic + Control . The logic component determines

the meaning of the algorithm whereas the control component only affects

its efficiency.

An algorithm may be formulated in different ways, producing same

behavior. One formulation, may have a clear statement in logic component

but employ a sophisticated problem solving strategy in the control

component. The other formulation, may have a complicated logic

component but employ a simple problem-solving strategy.

The efficiency of an algorithm can often be improved by improving the

control component without changing the logic of the algorithm and

therefore without changing the meaning of the algorithm.

The trend in databases is towards the separation of logic and control. The

programming languages today do not distinguish between them. The

programmer specifies both logic and control in a single language. The

execution mechanism exercises only the most rudimentary problem-solving

capabilities.

Computer programs will be more often correct, more easily improved, and

more readily adapted to new problems when programming languages

separate logic and control, and when execution mechanisms provide more

powerful problem-solving facilities of the kind provided by intelligent

theorem-proving systems.

77

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